Exercise I Consider the vector space R over R Let T R R be

Exercise (I): Consider the vector space R over R.

Let T : R R be a function defined by T(a1, a2, a3, . . .) = (0, a1, a2, a3, . . .) for all sequences (an) R .

(1) Prove that T is a linear transformation.

(2) Prove that T is one-to-one by showing that N(T) = {(0, 0, 0, . . .)}.

(3) Prove that T is NOT onto.

Solution

1)

a. T(a1+b1,a2+b2,a3+b3,....)=(0,a1+b1,a2+b2,...)=(0,a1,a2,...)+(0,b1,b2,...)=T(a1,a2,..)+T(b1,b2,..)

b. T(ca1,ca2,ca3,....)=(0,ca1,ca2,...)=c(0,a1,a2,...)=cT(a1,a2,..)

Hence T is linear

2.

T(a1,a2,a3,....)=(0,a1,a2,...)=(0,0,0,0,...)

Hence, a1=a2=a3=...=0

Hence, N(T)={(0,0,0,...)}

3)

T is not onto as it maps only to sequences of the form: (0,a1,a2,...)

eg. (1,0,0,0,...) is not in the image